Congress can't override substantive rules of constitutional law
Marbury v. Madison is a binding interpretation of what the U.S. Constitution permits or denies, and in substance, this law seeks to change that interpretation of the scope of the judicial power, so that interpretation may not be overruled except via a Constitutional amendment.
Neither the Supreme Court nor any lower federal court, under their appellate jurisdiction, will declare unconstitutional or otherwise
adjudicate unconstitutional any law passed by Congress; neither the
Supreme Court nor any lower federal court will hear or otherwise
engage in cases or controversies in which one or both parties put into
discussion the constitutionality of a law passed by Congress, or ask
for a law or a statue passed by Congress to be declared
The language in italics is jurisdiction stripping language, which I discuss below, and which is also discussed in another answer.
But, the language in bold is enunciating a substantive rule of law regarding how the judicial branch may resolve a case that is otherwise properly before it. And, Congress does not have the power to change that to make the U.S. Constitution a dead letter under its Article III jurisdiction regulation powers.
The language in bold language is a direct attempt to overrule a binding interpretation of the U.S. Constitution and that is beyond the authority of Congress to do, so the statute would be unconstitutional, at least, in part.
Yes, Congress can regulate the jurisdiction of the federal courts pursuant to Article III, Section 2 which states:
In all Cases affecting Ambassadors, other public Ministers and
Consuls, and those in which a State shall be Party, the supreme Court
shall have original Jurisdiction. In all the other Cases before
mentioned, the supreme Court shall have appellate Jurisdiction, both
as to Law and Fact, with such Exceptions, and under the Regulations as
the Congress shall make.
But, there are parts of Article III that apply in addition to the power of Congress to create "Exceptions" the appellate jurisdiction of the U.S. Supreme Court, and the power to create and modify the "inferior courts" that exist.
The first sentence of Article III, Section 1 of the U.S. Constitution states:
The judicial Power of the United States, shall be vested in one
supreme Court, and in such inferior Courts, as the Congress may from
time to time ordain and establish.
This is crucial, and interacts with the Exceptions power. The default provision is that all judicial power as defined in Article III, Section 2 is vested in the "supreme Court" unless and until that power is instead vested in an "inferior Court" established under Article III that Congress creates by law.
Therefore, Congress does not have the power to deny every court (or even every federal court) both original and appellate jurisdiction over any constitutionally justiciable claim arising under Article III, even if the claim is not within the express original jurisdiction of SCOTUS. If they deny every inferior Article III federal court jurisdiction over something within the constitutionally defined scope of the judicial power, then it reverts to the original jurisdiction of the U.S. Supreme Court even though it is not expressly made a part of the U.S. Supreme Court's original jurisdiction.
The judicial power of the federal courts collectively is defined in Article III, Section 2 of the U.S. Constitution and extends to all cases arising under the U.S. Constitution which would include a claim to have a provision of federal or state law declared unconstitutional as in violation of the constitution. It says:
The judicial Power shall extend to all Cases, in Law and Equity,
arising under this Constitution, the Laws of the United States, and
Treaties made, or which shall be made, under their authority; . . .
(This analysis is attributed to U.S. Supreme Court Justice Joseph Story.)
Now, this is not to say that Congress couldn't do something to make it harder procedurally to have statutes declared unconstitutional. For example, there would be a much harder claim of unconstitutionality if Congress vested original jurisdiction in all such cases in the United States in the U.S. District Court for the District of Wyoming, and then only assigned one judge to that district, and denied the U.S. Courts of Appeal or the U.S. Supreme Court, appellate jurisdiction over those decisions.
At some point, however, even this lesser restriction, rather than elimination of a judicial power would still be subject to challenge under the due process protections of the 5th Amendment.
Notably, Marbury v. Madison was a case brought in the original jurisdiction of the U.S. Supreme Court under a writ of mandamus, under the All Writs Act, and not in connection with its appellate jurisdiction.
So, Congress would also have to repeal or amend the "All Writs Act" to pull off the intent of the proposed statute, because the U.S. Supreme Court's original jurisdiction extends by statute to writs that are not appellate in nature even though this power is very rarely exercised.
A writ is a court order directed at a government official directing that government official to do something, or to refrain from doing something. But, there are many ways to back door a seemingly private cause of action, particularly one related to constitutionality, into a writ. And, if a court has jurisdiction over a writ, it has jurisdiction to entertain requests by litigants to have such writs issued.
Congress can't remove a state court forum
It is worth noting that every single state court from traffic court on up has concurrent jurisdiction with the federal courts to declare that a statute is unconstitutional, and that state courts frequently do declare state statutes to be unconstitutional.
Congressional jurisdiction to regulate jurisdiction is largely limited to regulation of the jurisdiction of the federal courts. It can put a federal question (e.g. copyright enforcement or disputes with the IRS) in the exclusive jurisdiction of the federal courts, but there are no cases in which Congress has been permitted to place a federal law in the exclusive jurisdiction of the federal courts while also denying any federal court jurisdiction over claims arising under that law. Otherwise, state court jurisdiction isn't regulated by Congress. And, the Constitution specifically requires all federal, state and local officials to swear to uphold the U.S. Constitution which arguably provides an independent basis for state court jurisdiction over constitutionality claims arising under the U.S. Constitution.
This is a really important point. For example, suppose that someone who lives in the same state as you do sues you entirely under state law in a state court, and that state's courts require you to bring any claim you have against that person in state court over which that state court has jurisdiction as a counterclaim or you forfeit that claim forever. If you have federal claims against the person who sued you in state court, and your claims are not one of the handful of issues (e.g. copyright enforcement) that are in the exclusive jurisdiction of the federal courts, you must enforce your federal claims against that person as counterclaims in that state court case, or you will lose them forever.
For example, suppose that your employer sues you in state court for conversion (i.e. stealing company property) and you have a right to sue the employer for not paying you the right amount for your overtime work under federal law. Then, you must bring your federal overtime claims in state court as counterclaims to the conversion action, rather than in federal court.
Similarly, even though state criminal charges are always brought in state courts, a criminal defendant in a state court criminal case, can raised arguments arising under the U.S. Constitution including a determination that a state criminal law is unconstitutional, in state court as a defense, even though the only federal court recourse a criminal defendant has is through an appeal to the U.S. Supreme Court or a post-conviction writ of habeas corpus brought in federal district court after all state direct appellate relief is exhausted, after petitioning to the U.S. Supreme Court, and after all state post-conviction relief (including petitioning the final state order to the U.S. Supreme Court) is exhausted. In practice, this means, criminal defendants have no meaningful access to the federal courts other than two petitions for certiorari to the U.S. Supreme Court which are discretionary, until they have been incarcerated wrongfully for five or ten years.
But, federal defenses can and routinely are raised in the state court trial (and indeed, federal defenses that could be raised in a state trial court may not be raised in a habeas corpus petition in federal court unless they were first raised in or before the original state court trial).
N.B.: Federal claims in the exclusive original jurisdiction of state courts
The extremes to which jurisdiction stripping is allowed are explored in the handful of claims arising under federal law that are expressly not within the scope of the jurisdiction of any federal trial court or intermediate appellate court, or within the express non-appellate jurisdiction of the U.S. Supreme Court.
The most notable of these are affirmative private individual civil lawsuits against offenders under the federal robocall and junk fax law (a.k.a. the Telephone Consumer Protection Act a.k.a. the TCPA a.k.a. 47 U.S.C. § 227), which do not not require a writ, which may only be brought in state court, subject to an ultimate appeal to the U.S. Supreme Court. But, the federal courts have exclusively jurisdiction over litigation many kinds of claims other than private civil actions arising under the TCPA.
This law is much less constitutionally concerning than the one proposed in the question, however, because while Congress can't repeal the U.S. Constitution, it doesn't have to pass a law giving private individuals a private cause of action when they receive robocalls or junk faxes at all. It could pass a law that was enforceable by the FCC alone, for example, and in the case of the TCPA, there are persons, including the FCC and regulated persons who want to challenge a regulation issued by the FCC, who are entitled to utilize the federal courts to enforce the TCPA or to dispute it.
For example, there is no private cause of action to enforce most federal criminal laws (as such, not just involving the same harm) with a civil lawsuit by the victim against the criminal, in either federal court or state court, but that is not unconstitutional. This is because federal criminal laws can be enforced by government prosecutors and defended against by private individuals, in Article III federal courts.
Also, even private causes of action under the TCPA are subject to ultimate U.S. Supreme Court appellate review, and the U.S. Supreme Court is an Article III federal court.